I do not find that the essential tremors affect my sleep. I also have restless leg syndrome which is another diagnosis that may or may not be related to my essential tremors. I have had RLS since I was a child, way before I even knew what it is called. It is also an internal phenomena that is difficult to describe unless you have ever had it. RLS creates an intense urge to move my legs to find comfort. I do not have it every night, but it seems to come in waves. I may have it for a week straight before it subsides until the next episode. It mainly happens at night, although I also have it sometimes when I sit in a meeting, or church, or standing in line at the grocery store. I find that leg exercises sometimes helps with that.
Our bodies are complicated pieces of equipment. Sometimes I have to give up on trying to find out why I am experiencing certain challenges, and just go straight for learning to cope with diet, exercise, or medication as a last resort. At this time, I am not on any medication for my essential tremors or my RLS.
Just as a side note, my 45 year old son also has RLS. His is even worse than mine, and he frequently has difficulty sleeping.